New Coordinate Systems for Axisymmetric Black Hole Collisions

The two–body problem is a milestone in numerical relativity that has not yet been attained. Even the relatively simple case of the head–on collision of two equal mass, non-rotating black holes remains uncertain for black holes that are initially separated by large distances. By exploiting the 2–dimensional nature of the problem, the obvious first choice of coordinates in which to solve the Einstein equations for the axisymmetric collision of two black holes are the cylindrical ones. However, these coordinates suffer from a code–crashing axis instability that can be suppressed (somewhat) by adding a shift vector to maintain a diagonal metric, but at the expense of introducing steep shear features in the solutions that cannot be maintained in a stable manner. An alternative approach, developed more than 20 years ago, is to perform the evolutions using curvilinear Čadež coordinates. However, these evolutions are also highly unstable due to the saddle–point singularity in the coordinate system. Two methods that have proven to be more successful invoke elements of both the cylindrical and Čadež coordinates. These evolutions run well when the black holes are restricted to separations less than about 13M , where M is the single black hole mass. However, even the most accurate of these methods is only a partial success when applied to the Misner initial data since the code cannot evolve data with large separation parameters. In this paper, we present an alternative approach to the axisymmetric binary black hole collision by constructing two new classes of orthogonal body–fitting coordinate systems, specially adapted to the 2–body problem.